In the given figure, two circles touch each other at a point CC. Prove that the common tangent to the circles at CC bisects the common tangent at the points PP and QQ.
C A B P Q R


Answer:


Step by Step Explanation:
  1. We see that PRPR and CRCR are the tangents drawn from an external point RR on the circle with center AA.
    Thus, PR=CR   (i)PR=CR   (i)

    Also, QRQR and CRCR are the tangents drawn from an external point RR on the circle with center BB.
    Thus, QR=CR   (ii)QR=CR   (ii)
  2. From eq (i)eq (i) and eq (ii)eq (ii), we get
    PR=QR   [Both are equal to CR]PR=QR   [Both are equal to CR]

    Therefore, RR is the midpoint of PQPQ.
  3. Thus, we can say that the common tangent to the circles at CC bisects the common tangent at the points PP and QQ.

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